1 00:00:00,000 --> 00:00:00,580 2 00:00:00,580 --> 00:00:07,310 We need to factor negative 4t squared minus 12t minus 9. 3 00:00:07,309 --> 00:00:09,289 And a good place to start is to say, well, are there any 4 00:00:09,289 --> 00:00:11,459 common factors for all of these terms? 5 00:00:11,460 --> 00:00:13,270 When you look at them, well these first two are divisible 6 00:00:13,269 --> 00:00:16,980 by 4, these last 2 are divisible by 3, but not all of 7 00:00:16,980 --> 00:00:18,920 them are divisible any one number. 8 00:00:18,920 --> 00:00:21,270 Will, but you could factor out a negative 1, but even if you 9 00:00:21,269 --> 00:00:23,550 factor out a negative 1-- so you say this is the same thing 10 00:00:23,550 --> 00:00:30,990 as negative 1, times positive 4t squared plus 12t plus 9-- 11 00:00:30,989 --> 00:00:34,049 you still end up with a non-one coefficient out here 12 00:00:34,049 --> 00:00:37,119 and on the second degree term, on the t squared term. 13 00:00:37,119 --> 00:00:39,199 So you might want to immediately 14 00:00:39,200 --> 00:00:40,109 start grouping this. 15 00:00:40,109 --> 00:00:43,829 And if you did factor it by grouping, it would work, you 16 00:00:43,829 --> 00:00:45,500 would get the right answer. 17 00:00:45,500 --> 00:00:51,549 But there is something about this equation that might pop 18 00:00:51,549 --> 00:00:55,079 out at you that might make it a little bit simpler to solve. 19 00:00:55,079 --> 00:01:00,539 And to understand that, let's take a little bit of a break 20 00:01:00,539 --> 00:01:02,679 here on the right hand side, and just think about what 21 00:01:02,679 --> 00:01:07,689 happens if you take a plus b times a plus b, if you just 22 00:01:07,689 --> 00:01:10,060 have a binomial squared. 23 00:01:10,060 --> 00:01:13,060 Well you have a times a, which is a squared. 24 00:01:13,060 --> 00:01:17,070 Then you have a times that b, which is plus ab. 25 00:01:17,069 --> 00:01:21,309 Then you have b times a, which is the same thing is ab. 26 00:01:21,310 --> 00:01:24,814 And then you have b times b, or you have b squared. 27 00:01:24,814 --> 00:01:28,359 And so if you add these middle two terms, right here, you're 28 00:01:28,359 --> 00:01:33,849 left with a squared plus 2ab plus b squared. 29 00:01:33,849 --> 00:01:36,149 This is the square of a binomial. 30 00:01:36,150 --> 00:01:40,990 Now, does this right here, does 4t squared plus 12t plus 31 00:01:40,989 --> 00:01:43,399 9 fit this pattern? 32 00:01:43,400 --> 00:01:46,020 Well the 4t squared is a squared. 33 00:01:46,019 --> 00:01:49,079 So this right here is a squared. 34 00:01:49,079 --> 00:01:51,179 If that is a squared right there, then what 35 00:01:51,180 --> 00:01:52,345 does a have to be? 36 00:01:52,344 --> 00:01:55,450 If this is a squared, then a would be equal to the square 37 00:01:55,450 --> 00:01:56,150 root of this. 38 00:01:56,150 --> 00:01:57,940 It would be 2t. 39 00:01:57,939 --> 00:02:00,099 And if this is b squared, let me do that 40 00:02:00,099 --> 00:02:01,939 in a different color. 41 00:02:01,939 --> 00:02:07,670 If this right here is b squared, if the 9 is b 42 00:02:07,670 --> 00:02:10,110 squared, right there, then that means that 43 00:02:10,110 --> 00:02:12,170 b is equal to 3. 44 00:02:12,169 --> 00:02:15,399 It's equal to the positive square root of the 9. 45 00:02:15,400 --> 00:02:18,430 Now, this number, right here-- and actually it doesn't have 46 00:02:18,430 --> 00:02:20,150 to just be equal to 3, it might have been 47 00:02:20,150 --> 00:02:21,430 negative 3 as well. 48 00:02:21,430 --> 00:02:23,240 It could be plus or minus 3. 49 00:02:23,240 --> 00:02:27,000 But this number here, is it 2 times ab? 50 00:02:27,000 --> 00:02:27,330 Right? 51 00:02:27,330 --> 00:02:29,650 That's the middle term that we care about. 52 00:02:29,650 --> 00:02:32,010 Is it 2 times ab? 53 00:02:32,009 --> 00:02:36,889 Well if we multiply 2t times 3, we get 6t. 54 00:02:36,889 --> 00:02:40,259 And then if we multiply that times 2, you get 12t. 55 00:02:40,259 --> 00:02:48,599 This right here, 12t, is equal to 2 times 2t times 3. 56 00:02:48,599 --> 00:02:50,960 It is 2 times ab. 57 00:02:50,960 --> 00:02:53,060 And if this was a negative 3, we would look to see if this 58 00:02:53,060 --> 00:02:56,009 was a negative 12, but this does work for positive 3. 59 00:02:56,009 --> 00:02:59,030 So this it does fit the pattern of a perfect square. 60 00:02:59,030 --> 00:03:05,060 This is a square of a binomial. 61 00:03:05,060 --> 00:03:07,590 So if you wanted to factor this-- the stuff on the 62 00:03:07,590 --> 00:03:09,939 inside, you still have that negative 1 out there, the 4t 63 00:03:09,939 --> 00:03:13,490 squared plus 12t plus 9-- you could immediately say, well 64 00:03:13,490 --> 00:03:18,879 that's going to be a plus b times a plus b. 65 00:03:18,879 --> 00:03:29,569 Or 2t plus 3 times 2t plus 3, or you could just say, it's 2t 66 00:03:29,569 --> 00:03:31,329 plus 3 squared. 67 00:03:31,330 --> 00:03:32,350 It fits this pattern. 68 00:03:32,349 --> 00:03:35,269 And, of course, you can't forget about this 69 00:03:35,270 --> 00:03:36,450 negative 1 out here. 70 00:03:36,449 --> 00:03:38,750 You could have also solved it by grouping, but this might be 71 00:03:38,750 --> 00:03:40,460 a quicker thing to recognize. 72 00:03:40,460 --> 00:03:41,820 This is a number squared. 73 00:03:41,819 --> 00:03:43,259 That's another number squared. 74 00:03:43,259 --> 00:03:45,599 If you take each of those numbers that you're squaring, 75 00:03:45,599 --> 00:03:47,909 take their product and multiply it by 2, you have 76 00:03:47,909 --> 00:03:48,650 that right there. 77 00:03:48,650 --> 00:03:51,330 So this is a perfect square. 78 00:03:51,330 --> 00:03:51,933